

A084950


Array of coefficients of denominator polynomials of the nth approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.


16



1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16, 40320, 35280, 5400, 200, 1, 362880, 322560, 52920, 2400, 25, 3628800, 3265920, 564480, 29400, 450, 1, 39916800, 36288000, 6531840, 376320, 7350, 36, 479001600, 439084800, 81648000, 5080320, 117600, 882, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

A factorial triangle, with row sums A001040(n+1), n >= 0.
Conjecture: also coefficient triangle of the denominators of the (nth convergents to) the continued fraction w/(1+w/(2+w/3+w/... This continued fraction converges to 0.697774657964... = BesselI(1,2)/BesselI(0,2) for w=1.  Wouter Meeussen, Aug 08 2010
For general w, Bill Gosper showed it equals n! 2F3((1/2n/2,n/2); (1,n,n); 4*w).  Wouter Meeussen, Jan 05 2013
From Wolfdieter Lang, Mar 02 2013: (Start)
The row length sequence of this array is 1 + floor(n/2) = A008619(n), n >= 0.
The continued fraction 0 + K_{k>=1}(x/k) = x/(1+x/(2+x/(3+... has nth approximation P(n,x)/Q(n,x). These polynomials satisfy the recurrence q(n,x) = n*q(n1,x) + x*q(n2,x), for q replaced by P or Q with inputs P(1,x) = 1, P(0,x) = 0 and Q(1,x) = 0 and Q(0,1) = 1. The present array provides the Qcoefficients: Q(n,x) = sum(a(n,m)*x^m, m=0 .. floor(n/2)), n >= 0. For the P(n,x)/x coefficients see the companion array A221913. This proves the first part of W. Meeussen's conjecture given above.
The solution with input q(1,x) = a and q(0,x) = b is then, due to linearity, q(a,b;n,x) = a*P(n,x) + b*Q(n,x). The motivation to look at the q(n,x) recurrence came from an emails from Gary Detlefs, who considered integer x and various inputs and gave explicit formulas.
This array coincides with the SWNE diagonals of the unsigned Laguerre polynomial coefficient triangle A021009.
The entries a(n,m) have a combinatorial interpretation in terms of certain socalled labeled Morse code polynomials using dots (length 1) and dashes (of length 2). a(n,m) is the number of possibilities to decorate the n positions 1,2,...,n with m dashes, m from {0, 1, ..., floor(n/2)}, and n2*m dots. A dot at position k has a weight k and each dash between two neighboring positions has a label x. a(n,m) is the sum of these labeled Morse codes with m dashes after the label x^m has been divided out. E.g., a(5,2) = 5 + 3 + 1 = 9 from the 3 codes: dash dash dot, dash dot dash, and dot dash dash, or (12)(34)5, (12)3(45) and 1(23)(45) with labels (which are in general multiplicative) 5*x^2, 3*x^2 and 1*x^2 , respectively. For the array of these labeled Morse code coefficients see A221915. See the Graham et al. reference, p. 302, on Euler's continuants and Morse code.
Row sums Q(n,1) = A001040(n+1), n >= 0. Alternating row sums Q(n,1) = A058797(n). (End)
For fixed x the limit of the continued fraction K_{k=1}^{infinity}(x/k) (see above) can be computed from the large order n behavior of Phat(n,x) and Q(n,x) given in the formula section in terms of Bessel functions. This follows with the wellknown large n behavior of BesselI and BesselK, as given, e.g., in the Sidi and Hoggan reference, eqs. (1.1) and (1.2). See also the book by Olver, ch. 10, 7, p. 374. This continued fraction converges for fixed x to sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).  Wolfdieter Lang, Mar 07 2013


REFERENCES

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; AddisonWesley, 1994.
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974 (1991 5th printing).


LINKS

G. C. Greubel, Rows, n=0..149 of triangle, flattened
Avram Sidi and Philip E. Hogan, Asymptotics of modified Bessel functions of high order. Int. J. of Pure and Appl. Maths. 71 No. 3 (2011) 481498


FORMULA

a(n, m) = (nm)!/m!*binomial(nm,m).  Wouter Meeussen, Aug 08 2010
From Wolfdieter Lang, Mar 02 2013: (Start)
Recurrence (short version): a(n,m) = n*a(n1,m) + a(n2,m1), n>=1, a(0,0) =1, a(n,1) = 0, a(n,m) = 0 if n < 2*m. From the recurrence for the Q(n,x) polynomials given in a comment above.
Recurrence (long version): a(n,m) = (2*(nm)1)*a(n1,m) + a(n2,m1)  (nm1)^2*a(n2,m), n >= 1, a(0,0) =1, a(n,1) = 0, a(n,m) =0 if n < 2*m. From the standard three term recurrence for the unsigned orthogonal Laguerre polynomials. This recurrence can be simplified to the preceding one, because of the explicit factorial formula given above which follows from the one for the Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = Lhat(nm,m), with the coefficients Lhat(n,m) = A021009(n,m) of the unsigned n!*L(n,x) Laguerre polynomials.
For the e.g.f.s of the column sequences see A021009 (here with different offset, which could be obtained by integration).
E.g.f. for row polynomials gQ(z,x) := sum(Q(n,x)*z^n, z=0 .. infty) = (I*Pi*sqrt(x)/sqrt(1z))*(BesselJ(1, 2*I*sqrt(x)*sqrt(1z))*BesselY(0, 2*I*sqrt(x))  BesselY(1, 2*I*sqrt(x)*sqrt(1z))*BesselJ(0,2*I*sqrt(x))), with the imaginary unit I = sqrt(1) and Bessel functions. (End)
The row polynomials are Q(n,x) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z)  BesselJ(0,z)*BesselY(n+1,z)) with z := I*2*sqrt(x), and the imaginary unit I. An alternative form is Q(n,x) = 2*(w/2)^(n+1)*(BesselI(0,w)*BesselK(n+1,w)  BesselK(0,w)*BesselI(n+1,w)*(1)^(n+1)) with w := 2*sqrt(x). See A221913 for the derivation based on AbramowitzStegun's Handbook.  Wolfdieter Lang, Mar 06 2013
limit(Q(n,x)/n!,n > infinity) = BesselI(0,2*sqrt(x)). See a comment on asymptotics above.  Wolfdieter Lang, Mar 07 2013


EXAMPLE

The irregular triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 ...
O: 1
1: 1
2: 2 1
3: 6 4
4: 24 18 1
5: 120 96 9
6: 720 600 72 1
7: 5040 4320 600 16
8: 40320 35280 5400 200 1
9: 362880 322560 52920 2400 25
10: 3628800 3265920 564480 29400 450 1
11: 39916800 36288000 6531840 376320 7350 36
12: 479001600 439084800 81648000 5080320 117600 882 1
...Reformatted and extended by Wolfdieter Lang, Mar 02 2013
E.g., to get row 7, multiply each term of row 6 by 7, then add the term NW of term in row 6: 5040 = (7)(720); 4320 = (7)(600) + 20; 600 = (7)(72) + 96; 16 = (7)(1) + 9. Thus row 7 = 5040 4320 600 16 with a sum of 9976 = a(7) of A001040.
The denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96w + 9w^2.  Wouter Meeussen, Aug 08 2010
From Wolfdieter Lang, Mar 02 2013: (Start)
Recurrence (short version): a(7,2) = 7*72 + 96 = 600.
Recurrence (long version): a(7,2) = (2*51)*72 + 96  (51)^2*9 = 600.
a(7,2) = binomial(5,2)*5!/2! = 10*3*4*5 = 600. (End)


MATHEMATICA

Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k, {k, 1, n}]]]], w], {n, 16}]; (* Wouter Meeussen, Aug 08 2010 *)
(* or equivalently: *)
Table[( (nm)!*Binomial[nm, m] )/m! , {n, 0, 15}, {m, 0, Floor[n/2]}] (* Wouter Meeussen, Aug 08 2010 *)
row[n_] := If[n == 0, 1, x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Denominator // CoefficientList[#, x] &];
row /@ Range[0, 12] // Flatten (* JeanFrançois Alcover, Oct 28 2019 *)


CROSSREFS

Cf. A001040, A180047, A180048, A180049.
Cf. A021009 (Laguerre triangle). For the Anumbers of the column sequences see the Cf. section of A021009. A221913.
Cf. A052119.
Sequence in context: A005299 A185586 A128728 * A180317 A066654 A145960
Adjacent sequences: A084947 A084948 A084949 * A084951 A084952 A084953


KEYWORD

tabf,nonn,easy


AUTHOR

Gary W. Adamson, Jun 14 2003


EXTENSIONS

Rows 12 to 17 added based on formula by Wouter Meeussen, Aug 08 2010
Name changed by Wolfdieter Lang, Mar 02 2013


STATUS

approved



